Hidden Markov Models: Missing data and multiple data streams
Ron Togunov & Marie Auger-Méthé
1 Tutorial goals and set up
1.1 Objectives
Our main objectives are to handle some of the common data problems encountered with marine fastloc GPS data and diving data, using a narwhal dataset provided by Dr. Marianne Marcoux. Specifically, we aim to address the following:
Irregular locations
Time gaps
Including diving covariates
2 Loading the required packages
First, we’ll setup the workspace with required packages.
library(momentuHMM)
library(dplyr)
library(tidyr)
library(lubridate)
library(adehabitatLT)
library(sf)
library(tmap)
library(terra)
library(units)
library(stringr)
library(diveMove)
library(conicfit)
library(car)
library(mitools)
library(doFuture)
library(corrplot) # Pearson's correlation matrix using cor()Please also set working directory to “Day2” of the HMM workshop:
3 Import data and initial data processing
For simplicity, we will also only look at the data for the month of August, 2017.
tracks <- read.csv("data/tracks.csv") %>%
filter(!is.na(x) & !is.na(y)) %>% # remove missing locations
mutate(
time = ymd_hms(time), # define time
loc_class = factor(loc_class, # define location class factor levels
levels = c("GPS", 3, 2, 1, 0, "A", "B"))) %>%
# remove identical records
filter(!(time == lag(time) & x == lag(x) & y == lag(y) & loc_class == lag(loc_class)),
# filter only the month of august
month(time) == 8, day(time) > 7, day(time) <= 14)The classic HMM assumes the observation data is in discrete time and that there is no missing data in the predictor variables.
# Calculate time difference between locations
tracks <- tracks %>%
mutate(dt = ifelse(ID == lead(ID), # If next data row is same individual
difftime(lead(time), time, units = "mins"), NA
)) # calculate time difference
# Look at the frequency of time differences
hist(tracks$dt, 1000, main = NA, xlab = "Time difference (min)")
Next, we’ll convert the data to a spatial dataset using the
sf package and plot the data. First, we define the
coordinate reference system of the original data (in this case WGS84,
which is defined by the EPSG code 4326). Next, we will
project the data into NAD83(CSRS) UTM zone 21N (EPSG:2962), which will
projected the coordinates in meter units with minimal distortion for
this data set.
tracks <- tracks %>%
st_as_sf(coords = c("x", "y")) %>% # converts to an sf object
st_set_crs(4326) %>% # define CRS
st_transform(2962) # reproject data to a UTMFor the first part of this tutorial, we’ll use only the fastloc GPS data so we don’t have to deal with location error.
We lose some data, particularly near the end of the tracks, but we will integrate ARGOS locations later in this tutorial.
Now, we can map the data using the tmap package to
visualize what it looks like.
# plot GPS
tracks_gps %>%
group_by(ID) %>%
summarise(do_union = FALSE) %>%
st_cast("LINESTRING") %>%
tm_shape() +
tm_lines(col = "ID", palette = "Dark2")
3.1 Selecting a time interval for the HMM
There are two key decisions we must make, the temporal resolution to use, and how to address data gaps. The desired resolution depends predominantly on the biological question you are asking as different behaviours and biological processes occur at different spatial and temporal scales (e.g. seasonal migration, daily movement between foraging and resting grounds, and fine scale foraging decisions). Generally, higher resolution data is preferred as it has more information, however it is possible to have too-high of a resolution wherein information from fine-scale variability drowns out the signal from coarse-scale patterns of interest (e.g., seasonal migration). In this case, we will be linking the movement data with high resolution (75 s) dive data to look at finer-scale behaviours (on the order of a few hours). My rule of thumb, is that you want 3-50 data points per behaviour. For behaviours spanning several hours, that roughly corresponds to a desired resolution between 2 min and 60 min.
Let’s see what resolutions may be possible in the data by looking at the most frequent time gaps.
##
## 10 11 1 9 12 2
## 299 109 101 75 72 56
# Zoom in on short intervals
hist(tracks_gps$dt, 1000, main = NA, xlab = "Time difference (min)", xlim = c(0,100))
We see that that the most frequent time gap is 10 min, followed by 11, 1, 9, and 12 min. We also see the majority of the gaps are < 60 min, however some are in excess of 600 min. Although it may be possible to use 1 or 2 min resolution data, we will have to address the data gaps. Frequent and large data gaps can be difficult to handle, especially as the number of missing data points approaches or exceeds the existing data; we really want to avoid this. Let’s examine the potential data structure at different resolutions for the different animals.
We first create a function that can calculate the proportion of missing locations we would have for a given resolution.
# Make function to estimate proportion of missing location
p_na <- function(t_0, t_max, n_loc, resolution) {
max_n_loc <- length(seq(t_0, t_max, by = as.difftime(resolution, units = "mins")))
n_NA <- max_n_loc - n_loc
n_NA / max_n_loc
}We can now use this function look at the proportion of NAs we would get with 1, 2, 5, 10, and 20 min resolution.
# summarise tracks
track_summary <-
tracks_gps %>% st_drop_geometry() %>%
# limit to core period with data
group_by(ID) %>%
filter(dt >= 1) %>% # remove duplicate time (just for this stage)
summarise(n_loc = n(), # number of locations
p_NA_1m = p_na(first(time), last(time), n_loc, 1), # 1 min
p_NA_2m = p_na(first(time), last(time), n_loc, 2), # 2 min
p_NA_5m = p_na(first(time), last(time), n_loc, 5), # 5 min
p_NA_10m =p_na(first(time), last(time), n_loc, 10), # 10 min
p_NA_20m =p_na(first(time), last(time), n_loc, 20)) # 20 min
track_summary## # A tibble: 3 × 7
## ID n_loc p_NA_1m p_NA_2m p_NA_5m p_NA_10m p_NA_20m
## <chr> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 T172062 485 0.950 0.900 0.751 0.501 0.00206
## 2 T172064 479 0.951 0.902 0.754 0.508 0.0164
## 3 T172066 491 0.949 0.899 0.747 0.493 -0.0124Here we see that despite the large number of 1 min time steps, at that resolution, >95 /% of the potential locations are missing. Even at the 10 min interval, > 50% of the locations would be missing. Very large data gaps that contribute to much of the missing locations can be excluded from the analysis, therefore, for this tutorial, I will use a 10 min resolution as a compromise between high-resolution information and good temporal coverage.
3.2 Handling data gaps
There are several ways to deal with data gaps, and I will address four 1. Interpolation (linear or statistical) 2. Nullification 3. Path segmentation
3.2.1 Approach 1. Interpolation
For large datasets with few and small gaps, the simplest approach to use linear interpolation. First, let’s identify the most likely minute we have data.
# which minute has the most data
tracks_gps %>%
st_drop_geometry() %>% # must convert back to data.frame
group_by(ID) %>%
summarise(minute = str_sub(minute(time),-1)) %>%
table()## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.## minute
## ID 0 1 2 3 4 5 6 7 8 9
## T172062 81 78 62 52 37 40 40 45 37 32
## T172064 77 68 71 67 32 32 40 32 31 38
## T172066 138 52 73 48 44 42 37 27 22 18It looks like for all three tracks, the most amount of locations fall on 0 min (i.e., 10, 20, 30, 40, 50, 60 min). Next, for each track, we will create a vector of times in which to estimate locations.
# convert tracks back to data.frame with xy coordinates
tracks_gps_ls <- tracks_gps %>%
mutate(x = st_coordinates(tracks_gps)[,"X"],
y = st_coordinates(tracks_gps)[,"Y"]) %>%
st_drop_geometry() %>%
split(.,.$ID) # split into list
# create full time series on which to estimate locations rounded to the nearest 10 min
tracks_gps_ls_time <- tracks_gps %>%
st_drop_geometry() %>% # convert to data.frame
group_by(ID) %>%
summarise(time = seq(round_date(first(time), "10 min"),
round_date(last(time), "10 min"),
by = 60*10)) %>%
split(.,.$ID) # split into list## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.Now, we can interpolate the locations for each track.
# function to create a data frame with approximated locations
approx_locs <- function(tracks, times){
data.frame(ID = times$ID,
time = times$time,
x = approx(tracks$time, tracks$x,
xout = times$time)$y,
y = approx(tracks$time, tracks$y,
xout = times$time)$y)
}
# Interpolate the location at the times from the sequence
tracks_gps_linear <- mapply(approx_locs, tracks_gps_ls, tracks_gps_ls_time,
SIMPLIFY = F) %>%
do.call("rbind", .) # convert list of tracks back to a single data.frame
# remove row names added by rbind
rownames(tracks_gps_linear) <- NULL
# plot locations
plot(tracks_gps_linear$x, tracks_gps_linear$y, pch = 20, col = "red", xlab = "x", ylab = "y", asp = 1)
points(st_coordinates(tracks_gps)[,"X"], st_coordinates(tracks_gps)[,"Y"], pch = 20)
Looks like it works. Let’s try fitting an HMM to this. First, lets
prepare the data using prepData and plot the data to
estimate starting parameters.
# Setting up the starting values
mu0 <- c(50, 500) # Mean step length
sigma0 <- c(50, 500) # Sd of the step length
kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)Ok, were are ready. Let’s fit the HMM
set.seed(1)
# Fit a 2 state HMM
HMM_gps_linear <- fitHMM(prep_gps_linear, nbState = 2, dist = list(step = "gamma",
angle = "vm"), Par0 = list(step = c(mu0, sigma0), angle = kappa0), formula = ~ 1)
plot(HMM_gps_linear, ask = FALSE)## Decoding state sequence... DONE
That model does not look good. It’s clear from the mapped states that state 1 largely represents the original data, while state 2 primarily represents the interpolated data. This is a common problem when data gaps are frequent or large such that the information in the interpolated data outweighs the signal from the original observations. Generally, I would only use linear interpolation when data gaps are small (< 3 locations) or relatively infrequent (< 20 % of the modelled locations). In our data, some gaps are > 5 hours (30 locations) and > 50% of the modelled locations are interpolated. So we need to use another approach.
A slightly better way to interpolate locations is to fit a correlated
random walk (CRW) model to the data and predict the most likely
locations. momentuHMM contains wrapper functions to
interpolate missing locations by fitting continuous-time CRW to the data
based on the crawl package by Devin Johnson and Josh
London. There are many options to fit the crawl model, and
a detailed tutorial for analysis with crawl is available
here: https://jmlondon.github.io/crawl-workshop/crawl-practical.html.
Let’s try to fit the most basic model using the wrapper function
crawlWrap.
set.seed(1) # crawl often fails, so I recommend always setting a seed
# fit crawl
crw_gps_10 <- crawlWrap(obsData = tracks_gps, timeStep = "10 mins")## Fitting 3 track(s) using crawl::crwMLE...
## Individual T172062...## Beginning SANN initialization ...## Beginning likelihood optimization ...## Individual T172064...## Beginning SANN initialization ...
## Beginning likelihood optimization ...## Individual T172066...## Beginning SANN initialization ...
## Beginning likelihood optimization ...## Warning in sqrt(diag(Cmat)): NaNs produced## DONE## Warning in eval(c.expr, envir = args, enclos = envir): crawl::crwMLE for
## individual T172066 has NaN variance estimate(s)## Predicting locations (and uncertainty) at 10 mins time steps for 3 track(s) using crawl::crwPredict...## Warning in nax != nay: longer object length is not a multiple of shorter object
## length## Warning in nax != nay: longer object length is not a multiple of shorter object
## length
## Warning in nax != nay: longer object length is not a multiple of shorter object
## length## DONE
Notice how the predicted tracks do not make perfectly straight lines through missing sections (particularly noticeable in T172062). Next, we will extract the predicted locations and add them to the observed data.
# filter predicted locations
tracks_gps_crw <- data.frame(crw_gps_10$crwPredict) %>%
filter(locType == "p") %>%
dplyr::select(mu.x, mu.y, time,
ID) %>%
dplyr::rename(x = "mu.x", y = "mu.y")Now, let’s try to fit the same HMM as before on this data using the same starting parameters.
set.seed(1)
# prep data
prep_gps_crw <- prepData(tracks_gps_crw, type = "UTM")
# Setting up the starting values
mu0 <- c(50, 500) # Mean step length
sigma0 <- c(50, 500) # Sd of the step length
kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)
# Fit a 2 state HMM
HMM_gps_crw <- fitHMM(prep_gps_crw, nbState = 2, dist = list(step = "gamma",
angle = "vm"), Par0 = list(step = c(mu0, sigma0), angle = kappa0))
plot(HMM_gps_crw, ask = FALSE)## Decoding state sequence... DONE
That’s looking much better. It looks like state 1 represents a low-speed, high tortuosity resident state, while state 2 represents higher-speed, low tortuosity travelling state.
In many instances, this model may be sufficient. However, the significant proportion of interpolated locations used to fit the model is likely to affect our results. For example, the large interpolated gaps are still relatively straightened out and a very consistent speed, and may skew the definition of state 2 in particular. Simply interpreting the results with this issue in mind can be adequate when we only have few interpolated locations. Below, we will explore more formal ways to address this problem, which can be particularly useful when we have very large numbers of interpolated locations.
3.2.2 Approach 2. Nullify data gaps
One strategy to address large data gaps is to nullify the data
streams (i.e., step length and turning angle) during moderate/large
interpolated gaps where we expect that the estimated movement has is
largely independent of the observed data before or after the gap. The
maximum size of a gap to allow depends on the frequency of the missing
data, frequency of locations, study species, and behaviours of interest.
In this case, I will nullify predicted locations in gaps larger than 60
min. First, we will identify which steps need to be nullified, then we
will prepare the data and nullify the estimated step and
angle data streams. We will do this again later in the
tutorial, so we will wrap it into a function called
prepData_NAGaps.
# define function to replace step and turn angle of large gaps with NA
NA_gaps <- function(tracks, times){
# rows where location is within a large gap
rows <- which(
rowSums(apply(times, 1, function(X, tracks){
dplyr::between(tracks,
as.POSIXct(X[1], tz = "UTC"),
as.POSIXct(X[2], tz = "UTC"))
}, tracks$time))==1)
tracks$step[rows] <- NA
tracks$angle[rows] <- NA
return(tracks)
}
# define function to identify and nullify gaps
prepData_NAGaps <- function(track_list, tracks_crw, res, max_gap, ...){
# for each ID, identify which rows have gaps >= max_gap
gaps_ls_rows <- lapply(track_list, function(x){
which(difftime(lead(x$time), x$time, units = "mins") >= max_gap)
})
# create sequence of times for each track from gaps >= 60 min
gap_ls <- mapply(FUN = function(track, gaps){
# identify start and end date of each gap
gaps_ls_srt_end <- list(start = ceiling_date(track$time[gaps], paste(res, "min")),
end = floor_date(track$time[gaps+1], paste(res, "min")))
# combine into single vector for each track
data.frame(start = gaps_ls_srt_end$start, end = gaps_ls_srt_end$end)
},
track_list, gaps_ls_rows, SIMPLIFY = F)
# prep data and list by ID
prep_tracks <- prepData(tracks_crw, ...) %>%
{split(., .$ID)}
# Interpolate the location at the times from the sequence
mapply(FUN = NA_gaps, prep_tracks, gap_ls,
SIMPLIFY = F) %>%
do.call("rbind", .) # convert list of tracks back to a single data.frame
}
prep_tracks_gps_crw_NAgaps <- prepData_NAGaps(tracks_gps_ls, tracks_gps_crw, 10, 60, type = "UTM")Now, let’s try to fit the same HMM as above to this data with large gaps nullified.
set.seed(1)
# Setting up the starting values
mu0 <- c(50, 500) # Mean step length
sigma0 <- c(50, 500) # Sd of the step length
kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)
# Fit a 2 state HMM
HMM_gps_crw_NAgaps <- fitHMM(prep_tracks_gps_crw_NAgaps, nbState = 2, dist = list(step = "gamma", angle = "vm"), Par0 = list(step = c(mu0, sigma0), angle = kappa0))
plot(HMM_gps_crw_NAgaps, ask = FALSE)## Decoding state sequence... DONE
Visually, the difference is subtle.
## $step
## state 1 state 2
## mean 258.1724 705.2750
## sd 158.3769 255.9088
##
## $angle
## state 1 state 2
## mean 0.000000 0.00000
## concentration 2.706698 23.60863## $step
## state 1 state 2
## mean 282.3941 757.6624
## sd 156.3310 248.9599
##
## $angle
## state 1 state 2
## mean 0.000000 0.00000
## concentration 2.476524 19.03249However, the estimated parameters are quite different whether you account for the large gaps in data. When you nullify large gaps, the mean step length for both states is higher, and the turn angle concentration parameters is lower for both states (i.e., more tortuous). The fact that the parameters change for both states, suggests that the large gaps skewed the parameterisation of both states.
3.2.3 Approach 3. Path segmentation
Another strategy to deal with larger gaps is to segment the tracks with a new individual ID. This may be particularly appropriate for gaps where we may reasonably expect that the the underlying states are effectively independent of one another. Specifically, we may ask, over what period of time does the behaviour of the animal affect the subsequent behaviour. In this case, we may expect that the behaviour of a narwhal depends on the behaviour over the proceeding several hours, however is independent after 24 hours. We can split the tracks for gaps larger than a predetermined threshold by iterating the ID column. We will not implement this approach in this tutorial, however, it can be done using the following code:
gap_thresh <- 3*60 # in hours (3h for illustration)
# gaps no more than 6h per segment
new_ID <- (tracks_gps$dt > gap_thresh | tracks_gps$ID != lag(tracks_gps$ID)) %>% # if dif.time > gap_thresh or new ID
replace_na(TRUE) %>% # replace first NA with ID = 1
cumsum() %>% # iterate ID
paste(tracks_gps$ID, ., sep = "_")
# then smaller gaps <= gap_thresh can be interpolated with crawlWrap3.3 Integrating dive data
Let’s import the dive data and explore it.
dives <- read.csv("data/dives.csv") %>%
mutate(time = ymd_hms(time)) %>%
filter(month(time) == 8, day(time) > 7, day(time) <= 14)
head(dives)## ID time depth dt
## 1 T172062 2017-08-08 00:00:00 1.0 1.25
## 2 T172062 2017-08-08 00:01:15 7.5 1.25
## 3 T172062 2017-08-08 00:02:30 11.0 1.25
## 4 T172062 2017-08-08 00:03:45 5.5 1.25
## 5 T172062 2017-08-08 00:05:00 3.5 1.25
## 6 T172062 2017-08-08 00:06:15 1.0 1.25##
## 1.25 61.25 121.25 181.25 241.25 301.25 361.25
## 18547 52 16 6 1 1 1It appears as though there are relatively few gaps, however the gaps that exist are relatively long (> 60 mins). Therefore, I suggest that we should regularize and nullify the dive data.
# regularise data
dives <- dives %>%
group_by(ID) %>%
# regularise time series by 1.25 min
summarise(time = seq(first(time), last(time), by = 1.25*60)) %>%
# merge regularised time with original dive
left_join(dives, by = c("ID", "time"))## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.Next, we have to summarise the time series into concrete data streams
that can be modelled in the HMM. We can use the diveMove
package to identify individual dives and calculate dive statistics. We
must first convert the depth data to the TDR class for each
whale. As we are working with multiple whales, we will use
lapply to apply the createTDR to each whale –
we will also use lapply for most of diveMove
functions.
dives_ls <- split(dives, dives$ID)
dive_TDR <- lapply(dives_ls, function(data){
createTDR(time = data$time, depth = data$depth, dtime = 1.25*60, file = "data/dives.csv")
})
# generate interactive plot
plotTDR(dive_TDR[[1]]) # note: try zooming in to part of the diveNext, we must use the calibrateDepth to calibrate the
depth data. This function identifies wet and dry periods (for animals
that haul out), applies a zero-offset correction (ZOC), and identifies
all dives in the record and their phases. ZOC can be done using either
the offset or filter method. In this case, we
will assume the depth data is accurate and does not require an offset.
We should also specify dive.thr, which represents the
threshold depth below which an underwater phase should be considered a
dive – in this case, we will set it at 8 m. There are many other
optional parameters to identify dives that we will not get into in this
tutorial (See vignette("diveMove") for details).
# calibrate TDR and identify dive phases
dive_TDR_calib <- lapply(X = dive_TDR, FUN = calibrateDepth, zoc.method = "offset", offset = 0, dive.thr = 8)## Record is truncated at the beginning and at the end
## 39 phases detected## 818 dives detected## Record is truncated at the beginning and at the end
## 75 phases detected## 484 dives detected## Record is truncated at the beginning and at the end
## 41 phases detected## 628 dives detectedIn the interactive plot, try zooming into one of the dives and hover mouse over plot to preview the phases identified by `calibrateDepth’. The phases identified correspond to descent (D), descent/bottom (DB), bottom (B), bottom/ascent (BA), ascent (A), and surface (X). We can plot a specific dives as follows:
Now, we can calculate summary statistics for each dive using the
function diveStats. There are many dive metrics that are
estimated, and which ones to retain are species, data, and question
specific. In this case, we will retain 8 from those calculated by
diveStats: dive time, bottom time, maximum depth, bottom
distance (measure of “wiggling while at the bottom), post-dive
duration.
# calculate dive stats and add dive.id to each dive
dive_sum <- lapply(dive_TDR_calib, function(data){
mutate(diveStats(data, depth.deriv = FALSE), dive.id = row_number()) %>%
dplyr::select(dive.id, divetim, botttim, maxdep, bottdist, postdive.dur)}) # select variables of interest
# add dive.id with depth data to each depth record
dives_ls <- mapply(function(TDR, dives){
mutate(TDR, dive.id = dives@dive.activity$dive.id)
}, TDR = dives_ls, dives = dive_TDR_calib, SIMPLIFY = F)
# join TDR data with dive summary data
dives_ls <- mapply(function(TDR, dive_sum){
left_join(TDR, dive_sum, by = "dive.id")
}, TDR = dives_ls, dive_sum = dive_sum, SIMPLIFY = F)
# convert dive_ls back to a data.frame
dives_df <- do.call(rbind, dives_ls)Next, we will replace any bottom time of a valid dive (dive.id > 0) to 75 s, since at least some time must be spent at the bottom. Then, we will calculate one additional metric as a proxy for dive shape; the ratio of bottom time to dive time. Dives where \(\leq20\%\) of the time is spent at the bottom represent V-shaped dives, U-shaped dives are represented when \(>20\) and \(\leq50\%\) is spent at the bottom, and square dives are represented when \(>50\%\) of the time is spent at the bottom.
# replace NA bottom time of valid dives
dives_df$botttim[dives_df$dive.id > 0 & is.na(dives_df$botttim)] <- 75
# calculate proportion time at bottom
dives_df <- dives_df %>%
mutate(propbott = botttim/divetim)
# remove "dives" with no duration
dives_df <- dives_df %>%
filter(!(dive.id > 0 & is.na(divetim)))The next issue is that the dive data is at a different temporal resolution (75 s) than the location data (10 min). There are two options to include both data streams in the same HMM. First, we can choose to implement a hierarchical HMM, however, this is more complicated, and will be covered in tomorrow’s tutorial. The second, which we will use here, is to summarise the depth/dive data to a 10 min resolution and include them as additional data streams with step length and turning angle.
dives_sum <- dives_df %>%
# round time to same interval as location data
mutate(time = floor_date(time, "10 min")) %>%
# group rows by time interval
group_by(ID, time) %>%
# summarise data
summarise(NA_t = sum(is.na(depth))*1.25,
surf_t = sum(dive.id == 0)*1.25,
mean_dep = ifelse(NA_t == 10, NA, mean(na.rm = T, depth)),
max_dep = ifelse(NA_t == 10, NA, max(na.rm = T, depth)),
dive_t = ifelse(NA_t == 10, NA, sum(na.rm = T, divetim)),
bott_t = ifelse(dive_t < 5, NA, sum(na.rm = T, botttim)),
prop_bott = ifelse(dive_t < 5, NA, max(na.rm = T, propbott)),
max_dive_dep = ifelse(dive_t < 5, NA, max(na.rm = T, maxdep)),
bott_dist = ifelse(dive_t < 5, NA, max(na.rm = T, bottdist)),
post_dive_dur = ifelse(dive_t < 5, NA, max(na.rm = T, postdive.dur))) %>%
# remove -Inf values (typically error)
filter(!is.infinite(dive_t) & !is.infinite(bott_dist))## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
## -Inf
## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
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## -Inf## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.## # A tibble: 6 × 12
## # Groups: ID [1]
## ID time NA_t surf_t mean_dep max_dep dive_t bott_t
## <chr> <dttm> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 T172062 2017-08-08 00:10:00 0 3.75 17.3 31 1875 375
## 2 T172062 2017-08-08 00:20:00 0 6.25 7.56 20 1875 1125
## 3 T172062 2017-08-08 00:30:00 0 0 109. 123 7200 4200
## 4 T172062 2017-08-08 00:40:00 0 5 20.9 88 2100 1200
## 5 T172062 2017-08-08 00:50:00 0 5 9.94 24.5 1275 450
## 6 T172062 2017-08-08 01:00:00 0 1.25 53.2 76.5 4200 1575
## # … with 4 more variables: prop_bott <dbl>, max_dive_dep <dbl>,
## # bott_dist <dbl>, post_dive_dur <dbl>Note, we used 2 different ifelse statements. First,
ifelse(NA_t == 10, ...) ensured that we only calculate mean
and maximum depth for periods where we have at least 1.25 min of depth
data. Second, ifelse(surf_t > 5, ...) ensured that we
only calculate dive metrics when at least half of the 10 min interval is
spent in a dive, otherwise the animal is assumed to be at the surface
and its dive metric is NA.
Any of the eight variables can be used as data streams in the HMM, however, including too many would significantly increase the number of parameters to estimate, and consequently computation time. Therefore, we must select which variables to use, which there are different several approaches: 1. Variable can be selected using expert on the species behaviour and research question. 2. We preferably want to avoid variable with a lot of missing data. 3. The data streams should exhibit variation and evidence of clustering or multi-modality that may be tied to the underlying behaviours. 4. Data streams with no variation or conversely are very noisy contain little information on underlying behaviour. 5. We want to avoid variables that are overdispersed and which would be difficult to describe using a statistical distribution. 6. Given that they are tied to autocorrelated behaviours, they too should exhibit some – but not too much – autocorrelation. 7. One of HMM’s key assumptions that the data stream are be independent of each other, therefore, we should avoid select highly co-linear data streams.
Assuming all the variables are biologically relevant, let’s look at
structure in the data. I will also add a subjective score based on the
results (first number after # symbol).
vars <- c("mean_dep","max_dep","dive_t","bott_t","prop_bott",
"max_dive_dep","bott_dist","post_dive_dur")
# First, missing data
dives_sum %>%
summarise(mean_dep = sum(is.na(mean_dep)), # 2
max_dep = sum(is.na(max_dep)), # 2
dive_t = sum(is.na(dive_t)), # 2
bott_t = sum(is.na(bott_t)), # 1
prop_bott = sum(is.na(prop_bott)), # 1
max_dive_dep = sum(is.na(max_dive_dep)), # 1
bott_dist = sum(is.na(bott_dist)), # 1
post_dive_dur = sum(is.na(post_dive_dur))) # 1## # A tibble: 3 × 9
## ID mean_dep max_dep dive_t bott_t prop_bott max_dive_dep bott_dist
## <chr> <int> <int> <int> <int> <int> <int> <int>
## 1 T172062 120 120 120 216 216 216 216
## 2 T172064 366 366 366 476 476 476 476
## 3 T172066 210 210 210 359 359 359 359
## # … with 1 more variable: post_dive_dur <int># Second, evidence of multi-modality, balanced variation, and no over-dispersion
# as the dive variables are zero-bound, applying a log transformation makes it easier to see structure in the distirbution
# dives_sum %>%
# ungroup() %>%
# dplyr::select(vars) %>%
# apply(2, hist, 100)
hist(log(dives_sum$mean_dep), 100) # 2 good structure, high dispersion
## Warning in logit(dives_sum$prop_bott): proportions remapped to (0.025, 0.975)
# Third, balanced autocorrelation
dives_sum_filter <- filter(dives_sum, ID == "T172062")
acf(dives_sum_filter$mean_dep, na.action = na.pass) # 1 ACF~8 some variability
# subjective score
data.frame(var = c("mean_dep", "max_dep", "dive_t", "bott_t", "prop_bott",
"max_dive_dep", "bott_dist", "post_dive_dur"),
score = c(5, 6, 5, 1,
1, 4, 4, 1))## var score
## 1 mean_dep 5
## 2 max_dep 6
## 3 dive_t 5
## 4 bott_t 1
## 5 prop_bott 1
## 6 max_dive_dep 4
## 7 bott_dist 4
## 8 post_dive_dur 1From my subjective interpretation of these outputs, I think the five most promising variables to include are maximum depth, dive time, mean depth, maximum dive depth, and bottom distribution. Next, let’s merge the dive data streams the location data.
tracks_dives <- left_join(prep_tracks_gps_crw_NAgaps,
dives_sum[,c("ID", "time", "max_dep", "dive_t",
"mean_dep", "max_dive_dep", "bott_dist")],
by = c("ID", "time"))Now, we must check for co-linearity between the data streams to select 1–3 to use in the HMM. We can check co-linearity using the Pearson’s correlation matrix.
# calculate and plot check Paerson's correlation matrix
tracks_dives[,c("max_dep", "dive_t", "mean_dep", "max_dive_dep",
"bott_dist", "step", "angle")] %>%
na.omit() %>%
cor() %>%
corrplot(method="number")
step and angle seem quit independent from
all variables as does bott_dist. Unfortunately, there is
quite high correlation between the first four dive data streams, se we
will have to select which to use. dive_t and
mean_dep have the lowest correlation with
bott_dist and step, however
max_dep had the highest subjective score. For the purposes
of this tutorial, I will use max_dep and
bott_dist.
To get starting parameters, we can fit an HMM to each one independently. We will use the gamma distribution for both and for now will use the same starting parameters.
# identify whether there is 0 data
tracks_dives %>%
summarise(max_dep = sum(max_dep == 0, na.rm = T),
bott_dist = sum(bott_dist == 0, na.rm = T)) ## max_dep bott_dist
## 1 0 628# therefore, we need to include zero-mass parameters for bott_dist
# starting parameters (will use same ones for both for now)
mu0 <- c(130, 180) # mean
sigma0 <- c(60, 90) # sd
zm <- c(0.1, 0.1) # zero mass, where applicable
# fit dive-only HMMs
set.seed(1)
HMM_max_dep <- fitHMM(tracks_dives, nbStates = 2, dist = list(max_dep = "gamma"),
Par0 = list(max_dep = c(mu0, sigma0)))
HMM_bott_dist <- fitHMM(tracks_dives, nbStates = 2, dist = list(bott_dist = "gamma"),
Par0 = list(bott_dist = c(mu0, sigma0, zm)))Next, we can integrate these into one HMM together with step length and turning angle. When we specify the starting parameters, we want to think about how the states may look. For example, we might expect one resident state with slower movement, lower angular concentration, deeper dives, and greater at-depth variability. The second state may be travel, with faster movement, greater angular concentration, shallower dives, and less at-depth variability.
# prep model
stateNames = c("resident", "travel")
nbStates = length(stateNames)
dist = list(step = "gamma", angle = "vm", max_dep = "gamma", bott_dist = "gamma")
# Starting Pars
# view Pars from previous HMMs
getPar(HMM_gps_crw_NAgaps)$Par # state 1 ~ resident, state 2 ~ travel## $step
## [1] 282.3941 757.6624 156.3310 248.9599
##
## $angle
## [1] 2.476524 19.032493## $max_dep
## [1] 30.15317 407.61829 29.05073 163.55047## $bott_dist
## [1] 23.9091351 315.3866789 24.7190593 203.5173936 0.2223528 0.5215131 # therefore must switch parameter estimates in HMM_max_dep & HMM_bott_dist
# combine starting Pars
step0 <- getPar(HMM_gps_crw_NAgaps)$Par$step
angle0 <- getPar(HMM_gps_crw_NAgaps)$Par$angle
max_dep0 <- c(getPar(HMM_max_dep)$Par$max_dep[c(2,1)], # mu1, mu2
getPar(HMM_max_dep)$Par$max_dep[c(4,3)]) # sd1, sd2
bott_dist0 <- c(getPar(HMM_bott_dist)$Par$bott_dist[c(2,1)], # mu1, mu2
getPar(HMM_bott_dist)$Par$bott_dist[c(4,3)], # sd1, sd2
getPar(HMM_bott_dist)$Par$bott_dist[c(6,5)]) # zm1, zm2
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0, bott_dist = bott_dist0)Now, we can fit the HMM with movement and dive variables.
set.seed(1)
HMM_move_dive <- fitHMM(tracks_dives, nbStates=nbStates, stateNames=stateNames, dist=dist, Par0=Par0)Let’s see what it looks like.
## Decoding state sequence... DONE
## Computing pseudo-residuals...
The tracks look interesting. But there are some issues in the pseudo residuals.
Looking at the QQ plots the model appears to: - overestimate the
number of fast movement steps - underestimating higher turning angles -
underestimate shallow dives and over estimate deep dives - really wonky
description of bott_dist The autocorrelation functions
suggest there is remnant autocorrelation in step and
max_dep that are not well described by the model. Together,
this information suggests that there may be additional states that are
not represented. Let’s try to add one more state with an intermediate
speed, lower angle concentration, and shallow dives.
# define states
stateNames <- c("resident", "travel", "search")
nbStates <- length(stateNames)
# Starting Pars
# get Pars from last HMM_move_dive HMM
Pars <- getPar(HMM_move_dive)$Par
# combine starting Pars
step0 <- c(Pars$step[c(1,2)], mean(Pars$step[c(1,2)]), # mu
Pars$step[c(3,4)], mean(Pars$step[c(3,4)])) # sd
angle0 <- c(Pars$angle, 3)
max_dep0 <- c(Pars$max_dep[c(1,2)], 25, # mu
Pars$max_dep[c(3,4)], 10) # sd
bott_dist0 <- c(Pars$bott_dist[c(1,2)], mean(Pars$bott_dist[c(1,2)]), # mu
Pars$bott_dist[c(3,4)], mean(Pars$bott_dist[c(3,4)]), # sd
Pars$bott_dist[c(5,6)], mean(Pars$bott_dist[c(5,6)]))# zm
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0, bott_dist = bott_dist0)
# fit 3-state HMM
set.seed(1)
HMM_move_dive_3s <- fitHMM(tracks_dives, nbStates=nbStates, stateNames=stateNames, dist=dist, Par0=Par0)## Computing pseudo-residuals...
## Decoding state sequence... DONE
Interestingly, the step and angle QQ-plots
were not improved much, though the step ACF was improved.
However, compared to the 2-state model, there was a drastic improvement
in QQ-plot and ACF the max_dep data stream and marginal
improvement in the bott_dist data stream. The newly
described state appears to have very low step length, really wide
turning angle, minimal diving, and minimal at-depth activity, which may
actually be indicative of resting. The “resident” state has intermediate
speed and turning angle, but very deep dives and high at-depth activity,
suggesting it may represent foraging. Let’s try to fit one more 4-state
HMM, to try and address the remaining residuals: intermediate speed,
lower angle concentration, and more intermediate dives, which may
represent searching behaviour.
# define states
stateNames <- c("forage", "travel", "rest", "search")
nbStates <- length(stateNames)
# Starting Pars
# get Pars from last HMM_move_dive HMM
Pars <- getPar(HMM_move_dive_3s)$Par
# combine starting Pars
step0 <- c(Pars$step[c(1:3)], mean(Pars$step[3]), # mu
Pars$step[c(4:6)], mean(Pars$step[6])) # sd
angle0 <- c(Pars$angle, Pars$angle[3])
max_dep0 <- c(Pars$max_dep[c(1:2)], Pars$max_dep[3]/2, Pars$max_dep[3], # mu
Pars$max_dep[c(4:5)], Pars$max_dep[6]/2, Pars$max_dep[6]) # sd
bott_dist0 <- c(Pars$bott_dist[c(1:3)], Pars$bott_dist[3], # mu
Pars$bott_dist[c(4:6)], Pars$bott_dist[6], # sd
Pars$bott_dist[c(7:9)], Pars$bott_dist[9]) # zm
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0, bott_dist = bott_dist0)
# fit 4-state HM
set.seed(1)
HMM_move_dive_4s <- fitHMM(tracks_dives, nbStates=nbStates, stateNames=stateNames, dist=dist, Par0=Par0)## =======================================================================## Fitting HMM with 4 states and 4 data streams## -----------------------------------------------------------------------## step ~ gamma(mean=~1, sd=~1)## angle ~ vm(concentration=~1)## max_dep ~ gamma(mean=~1, sd=~1)## bott_dist ~ gamma(mean=~1, sd=~1, zeromass=~1)##
## Transition probability matrix formula: ~1##
## Initial distribution formula: ~1## =======================================================================## DONE## Computing pseudo-residuals...
## Decoding state sequence... DONE
The QQ-plot for step is quite a bit improved, as is for
bott_dist. The ACF for step is also quite
improved. At this point, I’m not sure the model can be significantly
improved with the existing data. Although HMMs with different number of
states generally shouldn’t be compared, since AIC generally favours more
states, it is a good sanity check.
## Model AIC
## 1 HMM_move_dive_4s 70831.17
## 2 HMM_move_dive_3s 71706.75
## 3 HMM_move_dive 73241.60–>
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